Chapter 4 · Ratios and rates
Identify the Model That Shows a Ratio
Learn to pick the model that shows a ratio. Name what the shaded and unshaded cells stand for, then tell a part-to-part ratio from a part-to-whole fraction.
You already know how to write a ratio, such as , to compare two quantities. In this lesson you learn to spot the picture that shows a ratio.
A model is a picture that stands for the numbers in a situation. Each model here is one shape cut into equal parts called cells. The shape can vary: you will see bars and a grid. Some cells are shaded and the rest are unshaded. All the cells together make the whole. In this lesson, the whole is always one collection made of two groups, and one cell stands for one item.
Picture a bag that holds 3 red marbles and 5 blue marbles, and nothing else. That is marbles in all. The bar below has 8 cells, one cell for each marble. The 3 shaded cells stand for the red marbles. The 5 unshaded cells stand for the blue marbles.
This one bar gives you two different comparisons. Do not mix them up, because they use different numbers.
- Part to part compares one group with the other group. Red marbles to blue marbles is shaded cells to unshaded cells, so the ratio is . This is called a part-to-part ratio.
- Part to whole compares one group with all the items. Red marbles to all the marbles is shaded cells to all cells, which is . This comparison matches the fraction of the bar that is shaded, .
Both comparisons start with 3, the red marbles. They end differently: 5 counts the other group, and 8 counts the whole.
The order of the words still gives the order of the numbers, just as when you write a ratio. Blue marbles to red marbles is unshaded cells to shaded cells, so that ratio is .
Key concept
Name the cells, then pick the comparison
A model with shaded and unshaded cells fits a ratio when the two groups together make up one whole and one cell stands for one item.
- Name the cells first. Say which group the shaded cells stand for and which group the unshaded cells stand for.
- A ratio of one group to the other group is part to part. It compares the shaded cells and the unshaded cells, in the order the words give.
- Shaded cells to all cells is part to whole. It matches the fraction of the whole that is shaded. It is a different comparison from the ratio between the two groups.
So when a ratio gives the counts of the two groups, the model that shows it has those two counts as its shaded cells and its unshaded cells.
Worked example
Match a ratio to a bar
One row at a school play has 4 filled seats and 3 empty seats, and no other seats. One cell stands for one seat. A shaded cell is a filled seat and an unshaded cell is an empty seat. Which bar shows the ratio of filled seats to empty seats, ? Bar A has 7 cells with 3 shaded. Bar B has 7 cells with 4 shaded.
Name the cells. Shaded cells are filled seats and unshaded cells are empty seats. Filled seats to empty seats compares one group with the other group, so it is a part-to-part ratio: shaded cells to unshaded cells. It is not shaded cells to all the cells, which is the part-to-whole comparison of filled seats with the whole row.
Work out what the right bar must have. It needs 4 shaded cells for the filled seats and 3 unshaded cells for the empty seats. The two groups together make up the whole row, so add them to find the number of cells in all.
Check Bar A. Every cell is either shaded or unshaded, so shaded cells plus unshaded cells equals all cells. Subtract the 3 shaded cells from all 7 cells to find the unshaded cells: there are 4. Bar A shows 3 filled seats to 4 empty seats, which is 3:4. The row has 4 filled seats and 3 empty seats, so reject Bar A.
Check Bar B the same way. Subtract its 4 shaded cells from all 7 cells: it has 3 unshaded cells. Bar B shows 4 filled seats to 3 empty seats, which is 4:3. That matches the row.
Answer
Bar B shows the ratio of filled seats to empty seats, , because its 4 shaded cells are the filled seats and its 3 unshaded cells are the empty seats.
| Bar | Shaded cells (red) | Unshaded cells (blue) | Red to blue: shaded to unshaded (part to part) | Red to all: shaded to all (part to whole) |
|---|---|---|---|---|
8 cells, 3 shaded | , the fraction | |||
5 cells, 3 shaded | , the fraction |
Worked example
Choose from three models
A class garden has 5 tomato plots and 7 bean plots, and no other plots. One cell stands for one plot. A shaded cell is a tomato plot and an unshaded cell is a bean plot. Which model shows the ratio of tomato plots to bean plots, ? Model A has 12 cells with 7 shaded. Model B has 7 cells with 5 shaded. Model C has 12 cells with 5 shaded.
Tomato plots to bean plots compares one group with the other group, so it is part to part: shaded cells to unshaded cells, not shaded cells to all cells. The right model needs 5 shaded cells and 7 unshaded cells. The two groups make up the whole garden, so add them to find the cells in all.
Check Model B. Its unshaded cells are all its cells minus its shaded cells, so it has only 2. It shows 5 tomato plots to 2 bean plots, which is 5:2. Its 5:7 is shaded cells to all cells, the part-to-whole fraction: 5 of its 7 cells are shaded. Reject Model B.
Check Model A the same way. It has 5 unshaded cells, so it shows 7 tomato plots to 5 bean plots, which is 7:5. The two numbers are swapped, so reject Model A.
Check Model C. It has 7 unshaded cells, so it shows 5 tomato plots to 7 bean plots, which is 5:7. That matches the garden. Model C is drawn below.
Answer
Model C shows the ratio of tomato plots to bean plots, , because its 5 shaded cells are the tomato plots and its 7 unshaded cells are the bean plots.
Here is the idea to keep. When a model shows one whole made of two groups, with one cell for each item, name the cells first: the shaded cells are one group and the unshaded cells are the other group. A ratio of one group to the other group is part to part, so match its two numbers to the shaded count and the unshaded count, in the order the words give. Shaded cells to all cells is a different comparison. It is part to whole, the fraction of the model that is shaded.
Next, you will write ratios from word problems.